If your looking for simple and inexpensive, take a look at the Phottix Ares set. The transmitter has a test button on it that you can fire the flash with. It’s single group, no-TTL, but only $46 on Amazon.
As a relatively new D7500 shooter, one thing I haven’t seen mentioned yet is the autofocus area mode. Some have mentioned setting the camera to continuous autofocus, which makes sense for fast moving kids, but that only controls how the camera focuses, not WHAT it’s trying to focus on. If the camera got set to single point, and the OP isn’t keeping the focus point (Red rectangle) in the viewfinder on the kids, there’s not much chance of them being in focus, no matter how fast the shutter is.
Thus, I’d suggest checking what area mode it’s in, and set it to Auto Area, if it’s not already, to let the camera figure it out. If you want to keep control of where the camera is focusing by using single point, then you need to make sure the focus area is on the kids when the shuttter is tripped.
To see if this could be part of the problem, turn on “Show Focus Point” in your viewer software, to see what the camera was focusing on for the blurry shots.
BTW, one advantage of AF-Single over continuous is that in AF-S, the camera won’t release the shutter unless focus lock has been achieved. The upside is SOMETHING should be in focus when the shutter trips. The downside is that it can cause you to miss shots, which is why continuous is probably better.
I have a D7500 as well that I bought last November. I have David’s book, and like it. Being new to DSLR’s, I found it very helpful. My other suggestion is to download Nikon’s ManualViewer App, and then download the D7500 manual inside it. It’s a great way to get the manual if you don’t have one, and to always have the manual with you. In fact, I don’t think I’ve ever cracked open the paper copy that came with the camera.
Lastly, if you want a good education on Nikon’s autofocus system, get Steve Perry’s “Secrets to the Nikon Auto Focus System” e-book online, at www.backcountygallery.com